We will develop a powerful computational framework to map disease effects on the brain. Many degenerative and developmental diseases affect the cerebral cortex, but its 3-dimensional geometry is so complex and variable across subjects that disease effects are hard to detect or compare across subjects and groups. Creating a new direction in the field of computational anatomy, we will build on revolutionary advances in the field of partial differential equations (PDEs) that allow geometric and statistical manipulation of surfaces. Extending these tools, we will build a general mathematical framework to analyze, compare, and process neuroimaging data represented on the cortical surface. New algorithms will detect and map disease effects on human brain structure, visualizing systematic deficits in gyral/sulcal patterns, cortical shape and tissue distribution, and gray matter thickness, all of which are sensitive indicators of disease. Using novel mathematics of implicit functions and mappings between manifolds, we will build tools that elastically transform cortical surfaces from multiple subjects to a common anatomic template, encoding complex differences in neuroanatomy across subjects (Aim 1). Adapting these PDEs to map the profile of gray matter thickness at the cortical surface (Aim 2), we will encode how brain structure varies in large populations, and chart how the brain changes dynamically with age, in health and disease. New surface-based signal processing methods based on Laplace-Beltrami filters and adaptive grids will optimize detection of subtle or diffuse effects on cortical structure and function (Aim 3). Specialized methods will track average, group-specific anatomic patterns in the cerebral cortex. Building on our recent findings, our algorithms will be tested for uncovering deficits in Alzheimer's disease, including brain changes over time (Aim 4). These tools will be designed to be applicable to any brain disease (schizophrenia, autism, bipolar disorder, drug addiction/alcoholism, and other developmental/psychiatric disorders). We will mathematically combine algorithms from computational anatomy, PDEs, pattern theory, random field theory, and harmonic maps, to detect disease effects on the brain with maximal power. Validated on unique MRI datasets, these tools will be made publicly available and will find immediate application in a range of neuroimaging collaborations: they will chart the dynamics and spatial profiles of disease and medication effects in whole human populations.